######################################################################
##Chomp3Rows: Save this file as Chomp3Rows. To use it, stay in the   #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read Chomp3Rows : <Enter>                          #
##Then follow the instructions given there                           #
##                                                                   #
##Written by Doron Zeilberger, Temple University ,                   #
#zeilberg@math.temple.edu.                                           # 
#######################################################################
 
#Created: June 8, 2000
#This version: June 8, 2000
#Chomp3Rows: A Maple package to study 3-Rowed 2D-Chomp
#It accopmpanies Doron Zeilberger's article:
#`Three-Rowed Chomp'
#Available from http://www.math.temple.edu/~zeilberg/
#Please report bugs to zeilberg@math.temple.edu
 
 
print(`This is Chomp3Rows`):
print(` A Maple package to study and play 3-Rowed 2D-Chomp`):
print(`In accopmanies  Doron Zeilberger's article `):
print(` Three-Rowed-Chomp`):
print(`Available from http://www.math.temple.edu/~zeilberg/`):
print(``):
 
print(`Created: June 22, 2000.`):
print(`This version: Sept. 1, 2000.`):
lprint(``):
print(`Written by Doron Zeilberger, zeilberg@math.temple.edu`):
lprint(``):
print(`Please report bugs to zeilberg@math.temple.edu`):
lprint(``):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://www.math.temple.edu/~zeilberg/`):
 print(`For a list of the MAIN procedures type: ezra(); , for help with`):
 print(`a specific procedure, type: ezra(procedure_name);`):
 print(`For a list of all procedures type: ezra1(); , for help with`):
 print(`a specific procedure, type: ezra(procedure_name); `):
 print(``):
ezra1:=proc()
if args=NULL then
 print(`Contains the following procedures: CIW, DoesFit, DoesFit1, DoesFit2 `):
 print(` DoesFit3 , DoesFit4 , GoodMovef `):
  print(` GuessSym, hag3, hag3a,HakhiKatanLo,HK,Ini,Losers `):
 print(`  NewLosers, PlayChomp `):
 print(` PlayChompP,PTable,`):
 print(` F, T115, Winners, WinningMove, WinningMove1, yafe,  `):
 print(``):
 print(` With help with any of the above, type: ezra(ProcName); `):
fi:
end:
 
ezra:=proc()
if args=NULL then
 print(`Contains the following procedures:  `):
  print(` GoodMovef, Losers,  PlayChomp, PlayChompP `):
 print(`  PTable, T115, Winners, WINNERS, WinningMove, WinningMove1  `):
 print(``):
fi:
 
 
if nops([args])=1 and op(1,[args])=CheckNegative then
print(`CheckNegative(f,x,y): Given a rational function of the form`):
print(`P(x,y)+Q(y)/(1-x), checks that all its power-series`):
print(`coefficient are non-positive`):
fi:
 
if nops([args])=1 and op(1,[args])=CIW then
print(`CIW(POS,a,b): Given a  position in 3-rowed-Chomp, `):
print(`finds the symbolic implied winners (with same number of 3s),`):
print(`using the generic`):
print(`symbols a, and b for non-negative integers`):
fi:
 
if nops([args])=1 and op(1,[args])=DoesFit1 then
print(`DoesFit1(CPOS,SPOS,a,b): Given a concrete position, CPOS,`):
print(`and a symbolic position, decides whether CPOS is a`):
print(`specialization of SPOS`):
fi:
 
if nops([args])=1 and op(1,[args])=DoesFit2 then
print(`DoesFit2(CPOS,SPOS,a,b): If DoesFit1(CPOS,SPOS,a,b)`):
print(`is true `):
print(` returns the substition in the succesful case `):
fi:
 
if nops([args])=1 and op(1,[args])=DoesFit3 then
print(`DoesFit3(SPOS1,SPOS2,a,b): Given an alleged `):
print(` symbolic winning position, SPOS1,`):
print(` and an alleged symbolic losing position, SPOS2 `):
print(` decides whether they intersect `):
fi:
 
if nops([args])=1 and op(1,[args])=DoesFit4 then
print(`DoesFit4(SetSPOS1,SetSPOS2,a,b): Given a set of alleged `):
print(` symbolic winning positions, SetSPOS1,`):
print(`and a set of alleged symbolic losing positions, SetSPOS2`):
print(`decides whether one of the former intersects one of the latter`):
fi:
 
if nops([args])=1 and op(1,[args])=GoodMovef then
print(`GoodMovef(POS): using a pre-comuted table `):
print(` T115, computed by procedure Losers,`):
print(` Employing the straightforward notation`):
print(`for a position, POS, [row1,row2,row3], finds`):
print(`a good move (in the same notation), if it exists`):
print(`otherwise (i.e. if POS is a losing position), returns 0`):
print(` row3 must be <= 115 `):
 
fi:
 
if nops([args])=1 and op(1,[args])=GuessSym then
print(`GuessSym(resh,a,L): guesses a symbolic form for a sequence`):
print(`of triples, from the first few terms in terms`):
print(`of a sequence of the form [c3,c2+a,c1]`):
print(`where we need at least L terms at the end`):
fi:
 
if nops([args])=1 and op(1,[args])=hag3 then
print(`hag3(POS,c0,a): makes the number of 3's c0 more`):
print(`and number of 2's (if they exist) c0 less, and`):
print(`if they don't the number of 1's c0 less`):
print(`otherwise just the number of 3's c0 more`):
fi:
 
if nops([args])=1 and op(1,[args])=hag3a then
print(`hag3a(POS,c0): Like hag3a(POS,c0,a), but to be`):
print(`used in Winners rather than in Ini, so it`):
print(`keeps track of the behavior of POS[2]`):
print(`POS[3] resp.`):
fi:
 
if nops([args])=1 and op(1,[args])=HakhiKatanLo then
print(`HakhiKatanLo(kv,a,b,k,K): Given a set of symbolic winners`):
print(`with the first component (#of 3's) fixed at k, expressed`):
print(`in terms of the symbols a, and b denoting generic`):
print(`non-negative integers, finds the smalles tripe [k,i,j]`):
print(`s.t. 2*i+j is as small as possible, and 2*i+j<=K`):
print(`if it can't find anything, it returns 0`):
fi:
 
if nops([args])=1 and op(1,[args])=Ini then
print(`Ini(T,k,a): given a table where T[i] is the`):
print(`set of symbolic losers with POS[1]=i, for`):
print(`i=1..k-1 finds`):
print(`all the set of symbolic winners with POS[1]=k`):
print(`a denotes a generic (symbolic) non-negative`):
print(` integers `):
fi:
 
if nops([args])=1 and op(1,[args])=Losers then
print(`Losers(a,k): The symbolic lists, in terms of`):
print(`a table T, such that T[i] is the list of`):
print(`symbolic losers with POS[1]=i`):
fi:
 
if nops([args])=1 and op(1,[args])=NewLosers then
print(`NewLosers(T,a,k) : Knowing the losers for`):
print(`3-rowed Chomp with POS[1]<k, given by the`):
print(`table T, finds the set of symbolic losers`):
print(`for 3-rowed positions for which POS[1]=k`):
fi:
 
 
if nops([args])=1 and op(1,[args])=PlayChomp then
print(`PlayChomp(POS): `):
print(`plays 3-row CHOMP, starting at position POS`):
print(`and the human is first-to-move. It uses pre-computed `):
print(`data, computed by procesures Losers `):
print(`It can only handle positions where the smallest row has `):
print(` length <= 115. However, the other two rows can be as large `):
print(` as you wish. For example try Chomp([2000,2000,30]`):
fi:
 
if nops([args])=1 and op(1,[args])=PlayChompP then
print(`PlayChompP(POS): Pictorial version of PlayChomp `):
print(`plays 3-row CHOMP, starting at position POS`):
print(`and the human is first-to-move. It uses pre-computed `):
print(`data, computed by procesures Losers `):
print(`It can only handle positions where the smallest row has `):
print(` length <= 115. However, the other two rows can be as large `):
print(` as you wish. For example try Chomp([20,20,20]`):
print(`POS[3] should be <=20 `):
fi:
 
if nops([args])=1 and op(1,[args])=PTable then
print(`PTable(k): for k<=115, describes the P positions`):
print(`in 3-rowed Chomp with positions (gamma,beta,alpha)`):
fi:
 
if nops([args])=1 and op(1,[args])=RF then
print(`RF(kv,a,b,x,y): given a set, kv,  of symbolic triples where`):
print(`the second componet features a and the third component`):
print(`features b, finds the generating function of the complement`):
print(`in terms of the variables x and y`):
fi:
 
if nops([args])=1 and op(1,[args])=T115 then
print(`T115(a,k) the table of pre-computed symbolic losers up to last row=k`):
print(`k must be an integer 1<=k<=115 , a is a symbol`):
fi:
 
if nops([args])=1 and op(1,[args])=Winners then
print(`Winners(a,b,k,T): The symbolic sets, in terms of`):
print(`a table S, such that S[i] is the set of`):
print(`symbolic winners with POS[1]=i in the form [winner,loser]`):
print(`T is the precomputed table of losers`):
fi:
 
if nops([args])=1 and op(1,[args])=WINNERS then
print(`WINNERS(a,b,k): The symbolic set of Winners, in terms of`):
print(`a table S, such that S[i] is the set of`):
print(`symbolic winners with POS[1]=i in the form [winner,loser]`):
print(`where loser denotes the "right move" `):
print(`and a and b are generic (symbolic) non-negative integers`):
fi:
 
if nops([args])=1 and op(1,[args])=WinningMove then
print(`WinningMove(a,k,T,POS): In 3-Rowed (2D) Chomp `):
print(`Given a symbol a, an integer k,`):
print(`a (pre-computed) Losers' table T, where T[i] `):
print(` is the set of symboalic losers`):
print(`with POS[1]=i, for i=1,..., k, and given a position, POS`):
print(`expressed in the conjugate notaion [#3's, #2's, #1's] `):
print(`finds a winning move, if it exists, otherwise returns 0`):
fi:
 
if nops([args])=1 and op(1,[args])=WinningMove1 then
print(`WinningMove1(POS): Given a position POS, finds a winning move,`):
print(`using pre-computed data. nops(POS)<=3 and POS[1]<=115`):
fi:
 
if nops([args])=1 and op(1,[args])=yafe then 
print(`yafe(resh,a): given a list of triples`):
print(`all with the`):
print(`same first component, and using the symbol a,`):
print(`that denotes outputs a list `):
print(`whose (i+1)^th entry is the POS[3] of the P position with`):
print(`POS[2]=i if the last entry is a *, then it goes for ever`):
print(`with the last entry`):
fi:
 
end:
 
 
 
#CheckNegative(f,x,y): Given a rational function of the form
#P(x,y)+Q(y)/(1-x), checks that all its power-series
#coefficient are non-positive
#if also returns false if the test can't be applied
CheckNegative:=proc(f,x,y)
local P,Q,P1,degx,degy,coe1,coe2,i:
Q:=expand(subs(x=1,normal((1-x)*f))):
 
if denom(normal(Q))<>1 then
RETURN(false):
fi:
 
P:=normal((f-Q/(1-x))):
if denom(P)<>1 then
RETURN(false):
fi:
P:=expand(P):
 
for i from 0 to degree(Q,y) do
if coeff(Q,y,i)>0 then
 RETURN(false):
fi:
od:
if type(P,`+`) then
for i from 1 to nops(P) do
P1:=op(i,P):
degx:=degree(P1,x):
degy:=degree(P1,y):
coe1:=coeff(coeff(P1,x,degx),y,degy):
 
if coe1>0 then
 
coe2:=coeff(Q,y,degy):
if coe1+coe2>0 then
RETURN(false):
fi:
 
fi:
od:
 
elif (type(P,`*`) or type(P,`^`) or (type(P,integer) and P<>0)) then
 
P1:=P:
degx:=degree(P1,x):
degy:=degree(P1,y):
coe1:=coeff(coeff(P1,x,degx),y,degy):
 
if coe1>0 then
 
coe2:=coeff(Q,y,degy):
if coe1+coe2>0 then
RETURN(false):
fi:
 
fi:
 
elif P=0 then
RETURN(true):
else
RETURN(false):
 
fi:
 
true:
end:
 
 
#CIW(POS,a,b): Given a  position in 3-rowed-Chomp, 
#finds the symbolic implied winners (with same number of 3s),
#using the generic
#symbols a, and b for non-negative integers
CIW:=proc(POS,a,b) local i:
if (POS[3]>0) then
{[POS[1],POS[2],POS[3]+b+1],
seq([POS[1],POS[2]+i,POS[3]-i],i=1..POS[3])}:
else
 if type(POS[2],integer) then
{[POS[1],POS[2]+a+1,POS[3]+b],
[POS[1],POS[2],POS[3]+b+1]}:
 else
  {[POS[1],POS[2]+1,POS[3]+b],
[POS[1],POS[2],POS[3]+b+1]}
 fi:
fi:
end:
 
 
#DoesFit(CPOS,SetSPOS,a,b): Given a concrete position, CPOS,
#and a set of symbolic positions, SetSPOS, decides whether CPOS is a
#specialization of one of the symbolic positions of SetSPOS
DoesFit:=proc(CPOS,SetSPOS,a,b)
local i:
for i from 1 to nops(SetSPOS) do
 
if DoesFit1(CPOS,SetSPOS[i],a,b) then
 RETURN(true):
fi:
 
od:
false:
 
end:
 
#DoesFit1(CPOS,SPOS,a,b): Given a concrete position, CPOS,
#and a symbolic position, decides whether CPOS is a
#specialization of SPOS
DoesFit1:=proc(CPOS,APOS,a,b)
local eq,i,var,a0,b0:
eq:={seq(CPOS[i]-APOS[i],i=1..nops(APOS))}:
var:=solve(eq,{a,b}):
a0:=subs(var,a):b0:=subs(var,b):
if var=NULL  then
RETURN(false):
fi:
 
if type(a0, integer) then
 if a0<0 then
   RETURN(false):
 fi:
fi:
 
if type(b0, integer) then
 if b0<0 then
   RETURN(false):
 fi:
fi:
 
true:
end:
 
 
 
#DoesFit2(CPOS,SPOS,a,b): If DoesFit1(CPOS,SPOS,a,b)
#is true
#returns the substition in the succesful case
DoesFit2:=proc(CPOS,APOS,a,b)
local eq,i,var,a0,b0:
eq:={seq(CPOS[i]-APOS[i],i=1..nops(APOS))}:
var:=solve(eq,{a,b}):
a0:=subs(var,a):b0:=subs(var,b):
if var=NULL  then
RETURN(false,0):
fi:
 
if type(a0, integer) then
 if a0<0 then
   RETURN(false,0):
 fi:
fi:
 
if type(b0, integer) then
 if b0<0 then
   RETURN(false,0):
 fi:
fi:
 
true,var:
end:
 
 
 
#DoesFit3(SPOS1,SPOS2,a,b): Given an alleged 
# symbolic winning position, SPOS1,
#and an alleged symbolic losing position, SPOS2
#decides whether they intersect
DoesFit3:=proc(SPOS1,SPOS2,a,b)
local eq,i,var,a0,b0,ap,ap0:
 
eq:={seq(SPOS1[i]-subs(a=ap,SPOS2[i]),i=1..nops(SPOS1))}:
var:=solve(eq,{a,b,ap}):
a0:=subs(var,a):b0:=subs(var,b):ap0:=subs(var,ap):
if var=NULL  then
RETURN(false):
fi:
 
if type(a0, integer) then
 if a0<0 then
   RETURN(false):
 fi:
fi:
 
if type(b0, integer) then
 if b0<0 then
   RETURN(false):
 fi:
fi:
 
if type(ap0, integer) then
 if ap0<0 then
   RETURN(false):
 fi:
fi:
 
true:
end:
 
 
#DoesFit4(SetSPOS1,SetSPOS2,a,b): Given a set of alleged 
# symbolic winning positions, SetSPOS1,
#and a set of alleged symbolic losing positions, SetSPOS2
#decides whether one of the former intersects one of the latter
DoesFit4:=proc(SetSPOS1,SetSPOS2,a,b)
local i,j:
for i from 1 to nops(SetSPOS1) do
for j from 1 to nops(SetSPOS2) do
 
if DoesFit3(SetSPOS1[i],SetSPOS2[j],a,b) then
RETURN(true):
fi:
od:
od:
false:
end:
 
#FindPos(f,K,x,y,k): given a (2,1)-homog polynomial of
#degree K in x,y finds the set of positive terms
#and attaches a k at the first component
FindPos:=proc(f,K,x,y,k)
local i,gu:
gu:={}:
for i from 0 to K do
if coeff(coeff(f,x,i),y,K-2*i)>0 then
 gu:=gu union {[k,i,K-2*i]}:
fi:
od:
gu:
 
end:
 
 
#GuessSym(resh,a,L): guesses a symbolic form for a sequence
#of triples, from the first few terms in terms
#of a sequence of the form [c3,c2+a,c1]
#where we need at least L terms at the end
GuessSym:=proc(resh,a,L)
local i:
 
 
for i from nops(resh) by -1 to 2 while 
(resh[i][1]= resh[i-1][1] and resh[i][3]= resh[i-1][3] and 
resh[i][2]-resh[i-1][2]=1)  do 
od:
 
if nops(resh)-i+1>L then
RETURN([op(1..i-1,resh),[resh[i][1],a+resh[i][2], resh[i][3]]]):
else
RETURN(resh):
fi:
 
end:
 
 
#hag3(POS,c0,a): Given a position, POS, in 3-rowed (2D) Chomp
#returns the set of positions
#obtained by making the number of 3's one more
#and number of 2's (if they exist) one less, and
#if they don't the number of 1's one less
#a and b are the generic symbols used in POS[2] and
#POS[3] resp.
hag3:=proc(POS,c0,a) local i:
 
if type(POS[2],integer) then
 
if c0<=POS[2] and POS[2]>0 then
RETURN({[POS[1]+c0,POS[2]-c0,POS[3]]}):
 
elif POS[2]=0 and c0<=POS[3] then
RETURN({seq([POS[1]+c0,i,POS[3]-i-c0],i=0..POS[3]-c0)}):
 
else
RETURN({}):
fi:
 
fi:
 
if coeff(POS[2],a,0)>=c0 then
RETURN({[POS[1]+c0,POS[2]-c0,POS[3]]}):
 
elif coeff(POS[2],a,0)<c0 and coeff(POS[2],a,0)>0 then
RETURN({[POS[1]+c0,a,POS[3]]}):
 
else
RETURN({[POS[1]+c0,POS[2],POS[3]]} union hag3([POS[1],0,POS[3]],c0,a) ):
fi:
 
 
end:
 
 
#hag3a(POS,c0,a): Given a position, POS, in 3-rowed (2D) Chomp
#returns the set of positions
#obtained by making the number of 3's one more
#and number of 2's (if they exist) one less, and
#if they don't the number of 1's one less
#a and b are the generic symbols used in POS[2] and
#POS[3] resp.
hag3a:=proc(POS,c0,a) local i:
 
if type(POS[2],integer) then
 
if c0<=POS[2] and POS[2]>0 then
RETURN({[POS[1]+c0,POS[2]-c0,POS[3]]}):
 
elif POS[2]=0 and c0<=POS[3] then
RETURN({seq([POS[1]+c0,i,POS[3]-i-c0],i=0..POS[3]-c0)}):
 
else
RETURN({}):
fi:
 
fi:
 
if coeff(POS[2],a,0)>0 then
RETURN({[POS[1]+c0,POS[2]-c0,POS[3]]}):
 
else
RETURN({[POS[1]+c0,POS[2]-c0,POS[3]]} union hag3([POS[1],0,POS[3]],c0,a) ):
fi:
 
 
end:
 
 
#HakhiKatanLo(kv,a,b,k,K): Given a set of symbolic winners
#with the first component (#of 3's) fixed at k, expressed
#in terms of the symbols a, and b denoting generic
#non-negative integers, finds the smalles triple [k,i,j]
#s.t. 2*i+j is as small as possible, and 2*i+j<=K
#if it can't find anything, it returns 0
HakhiKatanLo:=proc(kv,a,b,k,K)
local i,j,A:
 
for A from 0 to K do
for i from 0 to A/2 do
j:=A-2*i:
 
if not DoesFit([k,i,j],kv,a,b) then
RETURN([k,i,j]):
fi:
od:
od:
0:
end:
 
 
#HK(kv,a,b,k): Given a set of symbolic positions
#kv, phrased in terms of a and b, and an integer
#k for which all first components of kv are equal
#to returns the smallest triple that is not in the
#support of kv, if it exists, otherwise returns 0
HK:=proc(kv,a,b,k) local f,f1,x,y,t,lu,i,ld:
f:=RF(kv,a,b,x,y):
 
f:=normal(subs({x=t^2*x,y=t*y},f)):
ld:=ldegree(numer(f),t):
 
for i from ld do
f1:=taylor(f,t=0,i+1):
lu:=expand(coeff(f1,t,i)):
 lu:=FindPos(lu,i,x,y,k):
 if lu<>{} then
  RETURN(lu):
 fi:
od:
 
end:
 
 
#Ini(T,k,a): given a table where T[i] is the
#set of symbolic losers with POS[1]=i, for
#i=1..k-1 finds
#all the set of symbolic winners with POS[1]=k
#a denotes generic (symbolic) non-negative
#integers
Ini:=proc(T,k,a) 
local gu,c0,i,mu:
 
if k=1 then
RETURN({[1,a,1],[1,0,0]}):
fi:
 
gu:={[k,a,1]}:
for c0 from 1 to k-1 do
mu:=T[k-c0]:
 
for i from 1 to nops(mu) do
 gu:=gu union hag3(mu[i],c0,a):
od:
 
od:
gu:
end:
 
 
 
#Losers(a,k): The symbolic lists, in terms of
#a table T, such that T[i] is the list of
#symbolic losers with POS[1]=i for i=1..k
Losers:=proc(a,k) 
local T,i:
T:=[]: 
for i from 1 to k do 
T:=[op(T),NewLosers(T,a,i)]
od:
 
T:
end:
 
#LosersC(T1,a,k): Given the list T of losers up
#to nops(T), continues it to k
LosersC:=proc(T1,a,k) 
local T,i:
if k<nops(T1) then
ERROR(`bad input`):
fi:
 
T:=T1: 
for i from nops(T1)+1 to k do 
T:=[op(T),NewLosers(T,a,i)]
od:
 
T:
end:
 
 
 
#LosersDoubleCheck(a,k): Like Loses, but at the end double-checks
#The symbolic lists, in terms of
#a table T, such that T[i] is the list of
#symbolic losers with POS[1]=i for i=1..k
LosersDoubleCheck:=proc(a,k) 
local T,i,W,j,gu,lu,S,f:
 
for i from 1 to k do 
T[i]:=NewLosers(T,a,i):
od:
 
W:=Winners(a,b,k,T):
 
for i from 1 to k do
lu:=convert(T[i],set):
gu:={seq(W[i][j][1],j=1..nops(W[i]))}:
f:=RF(lu union gu,a,b,x,y):
 
if (not CheckNegative(f,x,y)) or DoesFit4(lu,gu,a,b) then
print(`The conjectured list of losers failed when the number`):
print(`of 3's (first component) is`, i):
print(`Up to then, it is rigorously proved `):
for j from 1 to i-1 do
S[j]:=T[j]:
od:
RETURN(S):
fi:
od:
 
T:
end:
 
 
#NewLosers(T,a,k) : Knowing the losers for
#3-rowed Chomp with POS[1]<k, given by the
#table T, finds the set of symbolic losers
#for 3-rowed positions for which POS[1]=k
#K is the "confidence level"
NewLosers:=proc(T,a,k) 
local Losers,Winners,i,lu,b,n,mua,W,f,x,y:
Losers:=[]:
Winners:=Ini(T,k,a):
 
lu:=HK(Winners,a,b,k):
while lu<>{} do
Losers:=[op(Losers),op(lu)]:
 
n:=nops(Losers):
 
if n>=2 and
Losers[n][1]-Losers[n-1][1]=0 and
Losers[n][2]-Losers[n-1][2]=1 and
Losers[n][3]-Losers[n-1][3]=0 then
mua:=[op(1..n-2,Losers),[Losers[n-1][1],Losers[n-1][2]+a,Losers[n-1][3]]]:
W:=Ini(T,k,a):
for i from 1 to nops(mua) do
 W:=W union CIW(mua[i],a,b):
od:
 
f:=RF(W union convert(mua,set),a,b,x,y):
 
if  CheckNegative(f,x,y) and not DoesFit4(Losers,W,a,b) then
RETURN(mua):
fi:
fi:
 
 
 
for i from 1 to nops(lu) do
 Winners:=Winners union CIW(lu[i],a,b):
od:
 
f:=RF(Winners union convert(Losers,set),a,b,x,y):
 
if  CheckNegative(f,x,y) and not DoesFit4(Losers,Winners,a,b) then
RETURN(Losers):
fi:
 
lu:=HK(Winners union convert(Losers,set),a,b,k):
 
 
od:
 
Losers:
end:
 
#PTable(k): for k<=115, describes the P positions
#in 3-rowed Chomp with positions (gamma,beta,alpha)
PTable:=proc(k) 
local i,T,a:
T:=T115(a,k):
print(`The following describe the P-positions`):
print(`When the number of 3's is i=1,2, ..`,k):
print(` ... means that the last entry repeats for ever`):
print(`The lists give the number of 1's `):
print(`For number of 2's=0,1,2, 3...`) :
for i from 1 to min(9,k) do
lprint(`i=`,i,`  :`, op(yafe(T[i],a))):
od:
 
for i from 10 to min(99,k) do
lprint(`i=`,i,` :`, op(yafe(T[i],a))):
od:
 
for i from 100 to min(115,k) do
lprint(`i=`,i,`:`, op(yafe(T[i],a))):
od:
 
end:
 
#RF(kv,a,b,x,y): given a set, kv,  of symbolic triples where
#the second componet features a and the third component
#features b, finds the generating function of the complement
#in terms of the variables x and y
RF:=proc(kv,a,b,x,y)
local f,i,gu,mu:
f:=1/(1-x)/(1-y):
 
for i from 1 to nops(kv) do
gu:=kv[i]:
mu:=x^coeff(gu[2],a,0)*y^coeff(gu[3],b,0):
if coeff(gu[2],a,1)=1 then
 mu:=mu/(1-x):
fi:
if coeff(gu[3],b,1)=1 then
 mu:=mu/(1-y):
fi:
 
f:=f-mu:
 
od:
 
normal(f):
 
end:
 
#Winners(a,b,k,T): The symbolic sets, in terms of
#a table S, such that S[i] is the set of
#symbolic winners with POS[1]=i in the form [winner,loser]
#T is the precomputed list  of symbolic losers
Winners:=proc(a,b,k,T) 
local i,i1,j,gu,Gu,mu,U,lu,r,k0:
option remember:
for i from 1 to k do
gu:=T[i]:
Gu:={}:
for j from 1 to nops(gu) do
mu:=gu[j]:
lu:=CIW(mu,a,b):
 
for r from 1 to nops(lu) do
 Gu:=Gu union {[lu[r],mu]}:
od:
 
od:
 
for i1 from 1 to i-1 do
gu:=T[i1]:
 
for j from 1 to nops(gu) do
mu:=gu[j]:
lu:=hag3a(mu,i-i1,a):
 
for r from 1 to nops(lu) do
 
if coeff(mu[2],a,1)=1 and coeff(lu[r][2],a,1)=0 then
 
Gu:=Gu union {[lu[r],subs(a=0,mu)]}:
 
else
 
if coeff(lu[r][2],a,1)=1 then
 k0:=coeff(lu[r][2],a,0):
 
 if k0>=0 then
    k0:=0:
 fi:
 
   Gu:=Gu union {subs(a=a-k0,[lu[r],mu])}:
 
else 
Gu:=Gu union {[lu[r],mu]}:
fi:
 
fi:
od:
 
od:
 
od:
 
Gu:=Gu union {[[i,a,1],[i+a,1]]}:
if i=1 then
Gu:=Gu union {[[1,0,0],[1]]}:
fi:
U[i]:=Gu:
od:
 
U:
end:
 
 
#WINNERS(a,b,k): The symbolic set of Winners, in terms of
#a table S, such that S[i] is the set of
#symbolic winners with POS[1]=i in the form [winner,loser]
#where loser denotes the `right move'
#and a and b are generic (symbolic) non-negative integers
WINNERS:=proc(a,b,k) Winners(a,b,k,Losers(a,k)) : end:
 
#WinningMove(a,k,T,POS): Given a symbol a, an integer k,
#a Losers' list T, where T[i] is the set of symbolic losers
#with POS[1]=i, for i=1,..., k, and given a position, POS
#finds a winning move, if it exists, otherwise returns 0
WinningMove:=proc(a,k,T,POS) 
local b, W,POS1,i,gu,mu:
W:=Winners(a,b,k,T) :
 
if nops(POS)=1 then
 if POS[1]=1 then
  RETURN(0):
 else
 RETURN([1]):
 fi:
fi:
 
if nops(POS)=2 then
 if POS[2]=1 then
  RETURN(0):
 elif POS[2]=0 and POS[1]>1 then
  RETURN([POS[1]-1,1]):
 elif POS[2]=0 and POS[1]=1 then
  RETURN([1]):
 else
  RETURN([POS[1],1]):
 fi:
fi:
 
POS1:=POS[1]:
 
gu:=T[POS1]:
 
if DoesFit(POS,gu,a,b) then
 RETURN(0):
fi:
 
gu:=W[POS1]:
 
for i from 1 to nops(gu) do
  mu:=DoesFit2(POS,gu[i][1],a,b):
 if mu[1] then
 RETURN(subs(mu[2],gu[i][2])):
 fi:
od:
 
ERROR(`The Losers table`, op(T),  ` is wrong `):
 
end:
 
#yafe(resh,a): given a list of triples
#all with the
#same first component, and using the symbol a,
#that denotes outputs a list 
#whose (i+1)^th entry is the POS[3] of the P position with
#POS[2]=i if the last entry is a *, then it goes for ever
#with the last entry
yafe:=proc(resh,a)
local mu,i,T,lu,lu1:
 
for i from 1 to nops(resh) do
T[resh[i][2]]:=resh[i][3]:
od:
mu:=[]:
for i from 1 to nops(resh)-1 do
mu:=[op(mu),T[i-1]]:
od:
lu:=op(nops(resh),resh)[2]:
lu1:=op(nops(resh),resh)[3]:
if coeff(lu,a,1)=1 then
mu:=[op(mu),lu1,lu1,`...`]:
else
mu:=[op(mu),T[nops(resh)-1]]:
fi:
mu:
end:
 
 
#WinningMove1(POS): Given a position POS, finds a winning move
WinningMove1:=proc(POS) 
local a,T:
if nops(POS)>3 or nops(POS)=3 and POS[1]>115 then
ERROR(`Can only handle 3-rowed positions with POS[1]<=115`):
fi:
 
 
T:=T115(a,POS[1]): 
WinningMove(a,POS[1],T,POS):
end:
 
 
 
####Section on playing the game
 
#GoodMovef(POS): Using the straightforward notation
#for a position, POS, [row1,row2,row3], finds
#a good move (in the same notation), if it exists
#otherwise (i.e. if POS is a losing position), returns 0
GoodMovef:=proc(POS) local POS1:
if nops(POS)=1 then
POS1:=POS:
 
elif nops(POS)=2 then
POS1:=[POS[2],POS[1]-POS[2]]:
 
elif nops(POS)>3 or POS[3]>115 then
 ERROR(`Can only handle 3-rowed positions with smallest part<=115`):
 
else 
 POS1:=[POS[3],POS[2]-POS[3],POS[1]-POS[2]]:
fi:
 
POS1:=WinningMove1(POS1):
 
if POS1=0 then
RETURN(0):
fi:
 
if nops(POS1)=3 then
RETURN([POS1[1]+POS1[2]+POS1[3],POS1[1]+POS1[2],POS1[1]]):
elif nops(POS1)=2 then
RETURN([POS1[1]+POS1[2],POS1[1]]):
elif nops(POS1)=1 then
RETURN(POS1):
else
ERROR(`Something is wrong`):
fi:
end:
 
#Chop(posit,cell):given a position posit (a list of non-increasing
#positive integers), finds the result of perfoming the 
#move of chomping at the cell cell
Chop:=proc(posit,cell) local i,j,posit1,j1: i:=cell[1]: j:=cell[2]:  
#if  i>posit[j] or (i=1 and j=1) then ERROR(`Wrong move`): fi: 
posit1:=[op(1..j-1,posit)]:
for j1 from j to nops(posit) while posit[j1]>=i do 
if i>1 then posit1:=[op(posit1),i-1]: fi:od:
[op(posit1),op(j1..nops(posit),posit)]: end:
 
 
#DrawPos(POS,X): draws the position POS
DrawPos:=proc(POS,X)
local i,j:
for i from 1 to nops(POS) do
lprint(seq(X,j=1..POS[i])):
od:
end:
 
#PlayChomp(POS): plays Chomp starting at position POS
#and the human is first-to-move: 
PlayChomp:=proc(POS)
local  POS1,i,j,cell,lu:
if nops(POS)>3 or ( nops(POS)=3 and POS[3]>115) then
ERROR(`Can only handle 3 rows with smallest row <= 115`):
fi:
 
if (nops(POS)=3 and not(POS[1]>=POS[2] and POS[2]>=POS[3] and POS[3]>=1) )
or (nops(POS)=2 and not(POS[1]>=POS[2] and  POS[2]>=1)) 
or (nops(POS)=1 and  not(POS[1]>=1)) then
print(`You should have a non-increasing list of length<=3 of `):
print(`positive integers, e.g. PlayChomp([4,4,3]); `):
ERROR(`Bad position`):
fi:
 
POS1:=POS:
print(`The starting position  is `):
print(POS1):
while POS1<>[1] do
 
print(` It is your turn `):
lu:=nops(POS1):
print(` Which Row?, type an integer between`,1,`and`,lu,`followed by;<CR>`):
 
j:=readstat():
if not (1<=j and j<=nops(POS1)) then
print(`you must pick an integer between`,1,`and`,nops(POS1)):
print(`Try again!`):
j:=readstat():
fi:
lu:=POS1[j]:
print(` Which Column? Type an integer between`,1,`and`,lu,`followed by;<CR>`):
i:=readstat():
 
if not (1<=i and i<=POS1[j])  then
print(`you must pick an integer between`,1,`and`,POS1[j]):
print(`Try again!`):
i:=readstat():
fi:
 
 
print(``):
cell:=[i,j]:
 
if cell=[1,1] then
print(`Now there is nothing left, you lost!`):
RETURN():
fi:
 
POS1:=Chop(POS1,cell):
 
print(`After your move, the new position is`):
print(POS1):
 
POS1:=GoodMovef(POS1):
if  POS1=0 then
print(`I can't do anything good, so I am resigning`):
RETURN():
fi:
 
print(`I just moved. After my move,  the position  is `):
print(POS1):
od:
 
if POS1=[1] then
print(`Now the position is`):
print(POS1):
print(`Yea for me!, I won!, better luck next time.`):
 
RETURN():
fi:
 
end:
 
 
#PlayChompP(POS): plays Chomp starting at position POS
#and the human is first-to-move: FAST VERSION 
PlayChompP:=proc(POS)
local  POS1,i,j,cell,X,lu:
if nops(POS)>3 or ( nops(POS)=3 and POS[3]>115) then
ERROR(`Can only handle 3 rows with smallest row <= 115`):
fi:
 
if (nops(POS)=3 and not(POS[1]>=POS[2] and POS[2]>=POS[3] and POS[3]>=1) )
or (nops(POS)=2 and not(POS[1]>=POS[2] and  POS[2]>=1)) 
or (nops(POS)=1 and  not(POS[1]>=1)) then
print(`You should have a non-increasing list of length<=3 of `):
print(`positive integers, e.g. PlayChomp([4,4,3]); `):
ERROR(`Bad position`):
fi:
 
if POS[nops(POS)]>20 then
ERROR(`Too large position to be drawn, use PlayChomp instead `):
fi:
 
POS1:=POS:
print(`The starting position  is `):
DrawPos(POS1,X):
while POS1<>[1] do
 
print(` It is your turn `):
lu:=nops(POS1):
print(` Which Row?, type an integer between`,1,`and`,lu,`followed by;<CR>`):
 
j:=readstat():
if not (1<=j and j<=nops(POS1)) then
print(`you must pick an integer between`,1,`and`,nops(POS1)):
print(`Try again!`):
j:=readstat():
fi:
lu:=POS1[j]:
print(` Which Column? Type an integer between`,1,`and`,lu,`followed by;<CR>`):
i:=readstat():
 
if not (1<=i and i<=POS1[j])  then
print(`you must pick an integer between`,1,`and`,POS1[j]):
print(`Try again!`):
i:=readstat():
fi:
 
 
print(``):
cell:=[i,j]:
 
if cell=[1,1] then
print(`Now there is nothing left, you lost!`):
RETURN():
fi:
 
POS1:=Chop(POS1,cell):
 
print(`After your move, the new position is`):
DrawPos(POS1,X):
 
POS1:=GoodMovef(POS1):
if  POS1=0 then
print(`I can't do anything good, so I am resigning`):
RETURN():
fi:
 
print(`I just moved. After my move,  the position  is `):
DrawPos(POS1,X):
od:
 
if POS1=[1] then
print(`Now the position is`):
DrawPos(POS1,X):
print(`Yea for me!, I won!, better luck next time.`):
 
RETURN():
fi:
 
end:
 
 
#T115(a,k) the precomuted table of symbolic losers up to last row=k
#k must be an integer 1<=k<=115 , a is a symbol
T115:=proc(a,k) local T:
if k>115 or k<1 then
ERROR(`1<=k<=115`):
fi:
T:=
[[[1, 0, 2], [1, 1, 0]], [[2, a, 2]], [[3, 0, 3], [3, 2, 0], [3, 1, 3]], [[4, 0
, 4], [4, 1, 4], [4, 3, 0], [4, 2, 4]], [[5, 0, 5], [5, 1, 3], [5, 2+a, 4]], [[
6, 0, 5], [6, 3, 0], [6, 1, 5], [6, 2, 5]], [[7, 0, 6], [7, 2, 3], [7, 1, 6], [
7, 3+a, 5]], [[8, 0, 7], [8, 1, 5], [8, 4, 0], [8, 2, 6], [8, 3, 6]], [[9, 0, 7
], [9, 2, 3], [9, 1, 7], [9, 3+a, 6]], [[10, 0, 8], [10, 4, 0], [10, 1, 8], [10
, 2, 8], [10, 3, 8]], [[11, 0, 9], [11, 1, 7], [11, 3, 3], [11, 2, 9], [11, 4,
8], [11, 7, 3], [11, 5, 8], [11, 6, 8], [11, 8+a, 7]], [[12, 0, 9], [12, 5, 0],
[12, 4, 3], [12, 1, 10], [12, 2, 8], [12, 3, 9]], [[13, 0, 11], [13, 1, 9], [13
, 2, 7], [13, 6, 0], [13, 3, 9], [13, 4, 9], [13, 5, 9]], [[14, 2, 7], [14, 0,
11], [14, 1, 11], [14, 5, 3], [14, 3, 10], [14, 4, 10], [14, 6+a, 9]], [[15, 0,
12], [15, 1, 10], [15, 5, 3], [15, 7, 0], [15, 2, 11], [15, 3, 11], [15, 4, 11]
, [15, 6, 10]], [[16, 0, 12], [16, 2, 8], [16, 1, 12], [16, 7, 0], [16, 3, 12],
[16, 4, 10], [16, 5, 11], [16, 6, 11]], [[17, 0, 13], [17, 2, 10], [17, 1, 13],
[17, 6, 3], [17, 3, 12], [17, 4, 12], [17, 5, 12], [17, 7+a, 11]], [[18, 0, 14]
, [18, 3, 8], [18, 2, 11], [18, 1, 14], [18, 8, 0], [18, 5, 11], [18, 4, 13], [
18, 6, 12], [18, 7, 12]], [[19, 0, 15], [19, 1, 13], [19, 4, 8], [19, 7, 3], [
19, 2, 14], [19, 3, 14], [19, 5, 13], [19, 6, 13], [19, 8+a, 12]], [[20, 0, 15]
, [20, 3, 10], [20, 1, 15], [20, 9, 0], [20, 2, 15], [20, 8, 3], [20, 4, 14], [
20, 5, 14], [20, 6, 14], [20, 7, 14]], [[21, 4, 8], [21, 0, 16], [21, 2, 13], [
21, 1, 16], [21, 10, 0], [21, 3, 15], [21, 5, 15], [21, 6, 13], [21, 7, 14], [
21, 8, 14], [21, 9, 14]], [[22, 0, 17], [22, 1, 15], [22, 4, 10], [22, 8, 3], [
22, 2, 16], [22, 3, 16], [22, 5, 15], [22, 6, 15], [22, 7, 15], [22, 9+a, 14]],
[[23, 0, 17], [23, 3, 12], [23, 5, 8], [23, 1, 17], [23, 10, 0], [23, 2, 17], [
23, 4, 16], [23, 6, 16], [23, 8, 13], [23, 7, 16], [23, 9, 15]], [[24, 0, 18],
[24, 4, 10], [24, 1, 18], [24, 9, 3], [24, 2, 18], [24, 7, 10], [24, 3, 18], [
24, 5, 17], [24, 6, 17], [24, 8, 16], [24, 10+a, 15]], [[25, 5, 8], [25, 0, 19]
, [25, 1, 17], [25, 3, 13], [25, 11, 0], [25, 2, 19], [25, 4, 18], [25, 7, 13],
[25, 6, 17], [25, 8, 16], [25, 9, 16], [25, 10, 16]], [[26, 0, 19], [26, 4, 13]
, [26, 9, 3], [26, 1, 20], [26, 2, 18], [26, 7, 8], [26, 3, 19], [26, 5, 18], [
26, 6, 18], [26, 8, 17], [26, 10+a, 16]], [[27, 0, 21], [27, 1, 19], [27, 6, 10
], [27, 11, 0], [27, 4, 15], [27, 2, 20], [27, 3, 20], [27, 5, 19], [27, 7, 18]
, [27, 8, 18], [27, 9, 18], [27, 10, 18]], [[28, 0, 21], [28, 3, 16], [28, 7, 8
], [28, 5, 13], [28, 10, 3], [28, 1, 21], [28, 2, 21], [28, 4, 20], [28, 6, 19]
, [28, 8, 19], [28, 9, 17], [28, 11+a, 17]], [[29, 0, 22], [29, 2, 19], [29, 5,
13], [29, 1, 22], [29, 7, 10], [29, 12, 0], [29, 11, 3], [29, 3, 21], [29, 4, 
21], [29, 6, 20], [29, 8, 19], [29, 9, 19], [29, 10, 19]], [[30, 0, 23], [30, 1
, 21], [30, 3, 17], [30, 7, 10], [30, 8, 8], [30, 2, 22], [30, 13, 0], [30, 4,
21], [30, 5, 21], [30, 6, 21], [30, 9, 20], [30, 10, 18], [30, 11, 19], [30, 12
, 19]], [[31, 0, 23], [31, 2, 19], [31, 8, 8], [31, 6, 13], [31, 1, 23], [31, 
12, 3], [31, 3, 22], [31, 4, 22], [31, 5, 22], [31, 7, 21], [31, 9, 20], [31, 
10, 20], [31, 11, 20], [31, 13+a, 19]], [[32, 3, 17], [32, 1, 22], [32, 0, 24],
[32, 8, 10], [32, 2, 23], [32, 12, 3], [32, 14, 0], [32, 5, 21], [32, 4, 23], [
32, 6, 22], [32, 7, 22], [32, 9, 21], [32, 10, 21], [32, 11, 21], [32, 13, 20]]
, [[33, 0, 24], [33, 6, 13], [33, 9, 8], [33, 1, 24], [33, 5, 17], [33, 2, 24],
[33, 14, 0], [33, 3, 24], [33, 4, 24], [33, 7, 22], [33, 8, 22], [33, 11, 20],
[33, 10, 22], [33, 12, 21], [33, 13, 21]], [[34, 0, 25], [34, 1, 23], [34, 8, 
10], [34, 3, 23], [34, 2, 25], [34, 13, 3], [34, 11, 10], [34, 4, 24], [34, 5,
24], [34, 6, 24], [34, 7, 24], [34, 9, 23], [34, 10, 23], [34, 12, 22], [34, 14
+a, 21]], [[35, 0, 25], [35, 3, 20], [35, 9, 8], [35, 7, 13], [35, 1, 26], [35,
2, 26], [35, 15, 0], [35, 5, 21], [35, 4, 25], [35, 6, 25], [35, 8, 24], [35, 
12, 18], [35, 10, 23], [35, 11, 23], [35, 13, 22], [35, 14, 22]], [[36, 2, 23],
[36, 0, 27], [36, 1, 25], [36, 13, 3], [36, 11, 8], [36, 9, 13], [36, 3, 26], [
36, 5, 23], [36, 4, 26], [36, 6, 25], [36, 7, 25], [36, 8, 25], [36, 10, 24], [
36, 12, 23], [36, 14+a, 22]], [[37, 0, 27], [37, 2, 23], [37, 1, 27], [37, 15,
0], [37, 6, 18], [37, 11, 10], [37, 3, 27], [37, 5, 24], [37, 4, 27], [37, 8, 
24], [37, 7, 26], [37, 9, 25], [37, 10, 25], [37, 12, 24], [37, 13, 24], [37, 
14, 24]], [[38, 0, 28], [38, 2, 25], [38, 5, 19], [38, 1, 28], [38, 14, 3], [38
, 12, 8], [38, 10, 13], [38, 4, 26], [38, 3, 28], [38, 7, 25], [38, 6, 27], [38
, 8, 26], [38, 9, 26], [38, 11, 25], [38, 13, 25], [38, 17, 20], [38, 15, 24],
[38, 16, 24], [38, 18+a, 23]], [[39, 5, 18], [39, 0, 29], [39, 3, 23], [39, 2,
26], [39, 1, 29], [39, 16, 0], [39, 12, 10], [39, 4, 27], [39, 6, 26], [39, 7,
27], [39, 9, 24], [39, 8, 27], [39, 10, 26], [39, 11, 26], [39, 13, 25], [39, 
14, 25], [39, 15, 25]], [[40, 4, 21], [40, 0, 30], [40, 2, 27], [40, 9, 13], [
40, 5, 22], [40, 1, 30], [40, 15, 3], [40, 13, 8], [40, 3, 29], [40, 6, 28], [
40, 8, 25], [40, 7, 28], [40, 10, 27], [40, 12, 24], [40, 11, 27], [40, 14, 26]
, [40, 16, 25], [40, 19, 20], [40, 17, 25], [40, 18, 25], [40, 20+a, 24]], [[41
, 2, 26], [41, 1, 29], [41, 0, 31], [41, 6, 20], [41, 11, 10], [41, 17, 0], [41
, 3, 29], [41, 4, 29], [41, 14, 10], [41, 5, 29], [41, 7, 28], [41, 8, 28], [41
, 9, 28], [41, 10, 28], [41, 12, 27], [41, 13, 27], [41, 15, 26], [41, 16, 26]]
, [[42, 1, 28], [42, 0, 31], [42, 12, 8], [42, 5, 22], [42, 7, 18], [42, 10, 13
], [42, 2, 30], [42, 16, 3], [42, 3, 30], [42, 4, 30], [42, 6, 29], [42, 8, 29]
, [42, 9, 29], [42, 13, 23], [42, 11, 28], [42, 14, 27], [42, 15, 27], [42, 20,
20], [42, 17, 26], [42, 18, 26], [42, 19, 26], [42, 21+a, 25]], [[43, 0, 31], [
43, 4, 24], [43, 6, 20], [43, 1, 31], [43, 2, 31], [43, 16, 3], [43, 11, 13], [
43, 18, 0], [43, 14, 8], [43, 3, 31], [43, 5, 30], [43, 7, 30], [43, 9, 27], [
43, 8, 30], [43, 10, 29], [43, 12, 28], [43, 13, 28], [43, 15, 27], [43, 17, 27
]], [[44, 7, 18], [44, 0, 32], [44, 1, 32], [44, 4, 27], [44, 9, 18], [44, 2, 
32], [44, 18, 0], [44, 13, 10], [44, 3, 32], [44, 5, 31], [44, 6, 31], [44, 8,
30], [44, 10, 29], [44, 11, 29], [44, 12, 29], [44, 14, 28], [44, 15, 28], [44,
16, 28], [44, 17, 28]], [[45, 6, 20], [45, 0, 33], [45, 2, 30], [45, 11, 13], [
45, 5, 25], [45, 1, 33], [45, 14, 8], [45, 17, 3], [45, 3, 32], [45, 4, 32], [
45, 7, 31], [45, 9, 28], [45, 8, 31], [45, 10, 30], [45, 12, 29], [45, 13, 29],
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, 68], [110, 31, 70], [110, 33, 69], [110, 35, 69], [110, 37, 66], [110, 36, 69
], [110, 38, 68], [110, 40, 67], [110, 41, 67], [110, 43, 66], [110, 45+a, 65]]
, [[111, 1, 78], [111, 0, 80], [111, 4, 73], [111, 6, 69], [111, 12, 57], [111,
2, 79], [111, 10, 63], [111, 3, 79], [111, 19, 48], [111, 22, 42], [111, 25, 38
], [111, 31, 26], [111, 35, 18], [111, 5, 79], [111, 28, 33], [111, 9, 74], [
111, 7, 78], [111, 46, 0], [111, 41, 10], [111, 37, 18], [111, 8, 78], [111, 11
, 77], [111, 13, 76], [111, 15, 72], [111, 14, 76], [111, 16, 75], [111, 17, 75
], [111, 18, 75], [111, 20, 74], [111, 21, 74], [111, 23, 73], [111, 24, 73], [
111, 26, 72], [111, 27, 72], [111, 29, 71], [111, 30, 71], [111, 32, 70], [111,
33, 70], [111, 34, 70], [111, 36, 69], [111, 38, 68], [111, 39, 68], [111, 40,
68], [111, 42, 67], [111, 43, 67], [111, 44, 67], [111, 45, 67]], [[112, 1, 77]
, [112, 0, 80], [112, 6, 70], [112, 15, 53], [112, 9, 65], [112, 3, 78], [112,
2, 80], [112, 21, 45], [112, 4, 80], [112, 5, 78], [112, 34, 20], [112, 25, 38]
, [112, 17, 55], [112, 30, 30], [112, 29, 33], [112, 39, 13], [112, 42, 8], [
112, 36, 20], [112, 8, 77], [112, 7, 79], [112, 45, 3], [112, 11, 76], [112, 10
, 78], [112, 12, 77], [112, 13, 77], [112, 14, 77], [112, 16, 76], [112, 18, 75
], [112, 19, 75], [112, 20, 75], [112, 22, 74], [112, 23, 74], [112, 24, 74], [
112, 26, 73], [112, 27, 73], [112, 28, 73], [112, 31, 71], [112, 32, 71], [112,
33, 71], [112, 35, 70], [112, 37, 69], [112, 38, 69], [112, 40, 68], [112, 41,
68], [112, 44, 66], [112, 43, 68], [112, 46+a, 66]], [[113, 0, 80], [113, 6, 69
], [113, 1, 81], [113, 2, 79], [113, 12, 61], [113, 14, 57], [113, 9, 68], [113
, 19, 48], [113, 4, 79], [113, 3, 81], [113, 23, 42], [113, 5, 80], [113, 31, 
28], [113, 29, 33], [113, 41, 10], [113, 37, 18], [113, 40, 13], [113, 7, 79],
[113, 33, 28], [113, 47, 0], [113, 8, 79], [113, 46, 3], [113, 10, 78], [113, 
11, 78], [113, 13, 78], [113, 17, 73], [113, 15, 77], [113, 16, 77], [113, 18,
76], [113, 20, 75], [113, 21, 75], [113, 22, 75], [113, 24, 74], [113, 25, 74],
[113, 26, 74], [113, 27, 74], [113, 28, 74], [113, 30, 73], [113, 32, 72], [113
, 34, 71], [113, 35, 71], [113, 36, 71], [113, 38, 70], [113, 39, 70], [113, 42
, 69], [113, 44, 66], [113, 43, 69], [113, 45, 68]], [[114, 5, 71], [114, 1, 80
], [114, 0, 82], [114, 3, 77], [114, 10, 64], [114, 13, 59], [114, 2, 82], [114
, 4, 79], [114, 17, 53], [114, 21, 45], [114, 32, 26], [114, 30, 30], [114, 26,
38], [114, 7, 78], [114, 6, 80], [114, 42, 8], [114, 31, 30], [114, 8, 79], [
114, 41, 13], [114, 48, 0], [114, 44, 8], [114, 9, 79], [114, 11, 78], [114, 12
, 79], [114, 16, 74], [114, 14, 78], [114, 15, 78], [114, 18, 76], [114, 19, 76
], [114, 20, 76], [114, 22, 75], [114, 23, 75], [114, 24, 75], [114, 25, 75], [
114, 27, 75], [114, 28, 73], [114, 29, 74], [114, 34, 70], [114, 33, 72], [114,
36, 70], [114, 35, 72], [114, 37, 71], [114, 38, 71], [114, 39, 71], [114, 40,
71], [114, 43, 69], [114, 45, 68], [114, 46, 68], [114, 47, 68]], [[115, 0, 82]
, [115, 4, 76], [115, 9, 66], [115, 1, 83], [115, 2, 81], [115, 7, 72], [115, 
14, 59], [115, 16, 55], [115, 3, 82], [115, 23, 42], [115, 21, 48], [115, 5, 81
], [115, 41, 10], [115, 32, 28], [115, 33, 26], [115, 30, 33], [115, 6, 81], [
115, 37, 20], [115, 46, 3], [115, 43, 10], [115, 8, 80], [115, 10, 79], [115, 
11, 80], [115, 12, 78], [115, 13, 79], [115, 15, 78], [115, 17, 77], [115, 18,
77], [115, 19, 77], [115, 20, 77], [115, 22, 76], [115, 24, 76], [115, 26, 73],
[115, 25, 76], [115, 27, 75], [115, 28, 75], [115, 29, 75], [115, 31, 74], [115
, 35, 71], [115, 34, 73], [115, 36, 72], [115, 39, 70], [115, 38, 72], [115, 40
, 71], [115, 42, 70], [115, 44, 69], [115, 45, 69], [115, 47+a, 68]]]
:
 
[op(1..k,T)]:
end:
 
####End of Section on playing the game